Question

The sum of radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area is \[1628c{{m}^{2}}\], Then find its volume.

Answer 1

Hint: First let us assume r and h to be the radius and height of the cylinder respectively. Now we are given that r + h = 37. We will use this equation in the formula for total surface area which is given by $2\pi r\left( h+r \right)$ to find the radius of the cylinder. Now using the radius we will find the height of the cylinder by substituting the radius in equation r + h = 37. Hence we have the values of radius and height of the cylinder. Now the volume of the cylinder is given by $\pi {{r}^{2}}h$

Complete step by step answer:
Now let us say r is the radius of the base of the cylinder and h is the height of the cylinder.

Now we are given that the sum of radius and height is 37cm.
Hence we know that $r+h=37$ .
Now we know that the formula to calculate the total surface area of cylinder is given by $2\pi r\left( h+r \right)$
Now we are given that the total surface area of the cylinder is 1628.
Hence we have $2\pi r\left( h+r \right)=1628$
Now substituting the value of h + r we get, $2\pi r\left( 37 \right)=1628$
$\begin{align}
  & \Rightarrow r=\dfrac{1628}{2\times 37\times \pi } \\
 & \Rightarrow r=\dfrac{22}{\pi } \\
\end{align}$
Now let us take the value of $\pi $ as $\dfrac{22}{7}$ Hence we get,
$\begin{align}
  & \Rightarrow r=\dfrac{22}{\dfrac{22}{7}} \\
 & \Rightarrow r=7 \\
\end{align}$
Now substituting the value of r in $r+h=37$ we get,
$\begin{align}
  & \Rightarrow 7+h=37 \\
 & \Rightarrow h=37-7 \\
 & \Rightarrow h=30 \\
\end{align}$
Hence we have the value of r is 7 and h is 30
Now we want to calculate the Volume of the cylinder.
We know that the volume of the cylinder is given by $\pi {{r}^{2}}h$
Hence on substituting the values if r, h and $\pi $ we get,
$\begin{align}
  & \Rightarrow V=\dfrac{22}{7}\times {{7}^{2}}\times 30 \\
 & \Rightarrow V=22\times 7\times 30 \\
 & \Rightarrow V=4620 \\
\end{align}$
Hence the Volume of the given cylinder is $4620c{{m}^{3}}$

Note: Note that when dealing with cylinders we have two surface areas. The curved surface area of a cylinder is given by $2\pi rh$ and the total surface area of the cylinder is given by $2\pi r\left( h+r \right)$ .
The total surface area is nothing but the curved surface area + the area of the base and top circles of the cylinder.